Crossing Numbers of Graphs with Rotation Systems

We show that computing the crossing number and the odd crossing number of a graph with a given rotation system is NP-complete. As a consequence we can show that many of the well-known crossing number notions are NP-complete even if restricted to cubic graphs (with or without rotation system). In particular, we can show that Tutte’s independent odd crossing number is NP-complete, and we obtain a new and simpler proof of Hliněný’s result that computing the crossing number of a cubic graph is NP-complete. We also consider the special case of multigraphs with rotation systems on a fixed number k of vertices. For k = 1 we give an O(m log m) algorithm, where m is the number of edges, and for loopless multigraphs on 2 vertices we present a linear time 2-approximation algorithm. In both cases there are interesting connections to edit-distance problems on (cyclic) strings. For larger k we show how to approximate the crossing number to within a factor of k+4 4 /5 in time O(mk log m) on a...

• Type: Journal
• Year: 2011
• Where: ALGORITHMICA
• Authors: Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic
• Keywords: ALGORITHMICA 2011, Artificial Intelligence, Cubic Graph, NP-complete, Rotation Systems